42 Chapter 2. The AdS/CFT correspondence
Low viscosity bound The phenomenologically most striking prediction of AdS/CFT is that the viscosityηto entropy densitysratio is incredibly small
η s = 1
4π. (2.98)
This bound is satisfied to leading order in1/Nc in all theories with gravity duals computed up to now 6. It was observed at the RHIC heavy-ion collider that the quark gluon plasma sup-posedly formed in this experiment has an extremely low viscosity (well below any viscosity measured before) numerically comparable with the AdS/CFT value. Most of the models used to analyze the RHIC data are consistent with ratios in a range ofη/s≈ 4/34π . . .4π2 [89, 90, e.g.].
This discovery was even celebrated as an experimental possibility of testing the AdS/CFT cor-respondence. One has to be careful though since no QCD-dual gravity theory has been dis-covered yet and thus one has to rely on the universality of the observables to be measured. In the context of these viscosity investigations many different backgrounds have been employed in order to find out what makes this bound so universal. All investigated gauge theories with gravity duals show this universal behavior no matter if one breaks conformal symmetry, su-persymmetry or if one introduces flavor or a finite chemical potential. It is still under lively investigation which principle is the origin of the viscosity universality.
In a series of papers [26, 28, 29, 31, 32, 34, 35, 9] an identification of hydrodynamic modes with gravity objects was achieved leading to a detailed gravity description of the hydrody-namics in a strongly coupled fluid. Recently this framework has been extended to second order hydrodynamics [91, 92, 93, 94]. Here also a correction of the widely used Mueller-Israel-Stewart theory is proposed based on gravity consistency arguments. It is well known that hydrodynamics violates causality. Mueller-Israel-Stewart theory is a relativistic gener-alization of second order hydrodynamics which the authors of [91, 92, 93, 94] claim to be incomplete.
D3/D7-setup A particularly promising setup is the D3/D7-brane configuration described in 2.3. Its gauge dual contains massive quarks and a chemical potential can be consistently introduced. Further it exhibits confinement and thus a first order phase transition of the fun-damental matter in the spectrum. We will study this particular system in most of this thesis.
The calculation of meson spectra [38] in this system was one of the first phenomenological applications of AdS/CFT. Also ratios of B-meson masses were recently given [95].
A topic under ongoing investigation is that of heavy-light mesons [95, 96, 97] modeled by strings spanning from one D7-brane to another after having separated the D7-branes from each other.
Recently the hadron multiplicities after hadronization of the final state in a particle-antiparticle annihilation [98] have been modelled to surprising accuracy (see also [99]).
Interesting effects such as mass shift analogous to the Stark effect and chiral symmetry breaking are also observed in gauge/gravity duals with flavor for which pure-gauge
Kalb-6Note, that a recent investigation [87] had claimed that higher derivative corrections violate the viscosity bound for a certain family of models. But the same authors also found these very theories to be inconsistent violating microcausality [88] supporting again the idea of the universality of this bound.
2.5. More Phenomenology from AdS/CFT 43
Ramond B fields are turned on in the background, into which a D7 brane probe is embed-ded [55, 100, 101].
QCD duals Although some aspects of the D3/D7-brane configuration mirror QCD quite well one main point of criticism is that the dual gauge theory has too much symmetry. Re-member that on the gauge theory side we have N = 2 supersymmetric Yang-Mills theory coupled toN = 4SYM and the conformal symmetry is broken if the quarks become massive by seperating the D3 from the D7-branes. Also a finite temperature, i.e. a black hole back-ground metric breaks conformal symmetry. In a different backback-ground, the Constable-Myers background all of the supersymmetry is broken and the theory turns out to be confining [83].
Also chiral symmetry can be broken separately by choosing the background given in [37].
Nevertheless, all these approaches only manage to break part of the symmetry. An explicit QCD-dual has not been found, yet.
A special QCD dual: Sakai-Sugimoto model The Sakai-Sugimoto model is an alter-native D4/D8 anti-D8 brane system withNc D4-branes and Nf pairs of D8/anti-D8-branes.
Here the D4-branes generate the geometry very much like the D3 branes do in D3/D7 setups and the D8 and anti-D8 branes are the flavor branes corresponding to the D7. Since this model is the second most studied model (after the D3/D7-setup) introducing fundamental matter, we discuss also a few technical points here. This setup features no quark masses but two distinct phase transitions corresponding to the chiral symmetry breaking and deconfinement transi-tion, respectively. Supersymmetry is explicitly broken. In contrast to the D3-setup, there is one extra-dimension x4 in the worldvolume of the gauge theory. In order to come down to four space-time dimensions this extra coordinate needs to be compactified. There is also a geometrical argument for this coordinate to be periodic: together with the ”radial” coordinate uit forms a cigar-shaped submanifold, which has a tip atu=uT. To avoid a singularity at this tip,x4needs to be periodic with period2πR. The metric of the background at low temperature is
ds2 = ( u
RD4)3/2(dt2+δijdxidxj+f(u)dx24) + (RD4
u )3/2(du2
f(u)+u2dΩ24) (2.99) Thex4-circle shrinks to zero atu =uΛ and theD8and their antibranes have nowhere to end thus staying connected. So the chiralU(Nf)L×U(Nf)Ris broken to a diagonalU(Nf)V in the low temperature phase.
At finite temperature there always exist two solutions of which one is preferred at low temperature and the other at high temperature. Connected to this an asymptotic symmetry among the two circles (time-direction and x4) exists. In the high temperature phasetandx4
interchange roles (thef(u)in the metric is shifted from one to the other), so that thex4-circle now does not shrink to zero, but thet-circle does. Chiral symmetry is restored as the flavor branes may be parallel now.
The biggest advantage of this model over the D3/D7-setup is that chiral symmetry breaking can be achieved quite naturally. On the contrary, the quark masses are not incorporated from the start but also arise dynamically. Mesons have also been studied in the Sakai-Sugimoto
44 Chapter 2. The AdS/CFT correspondence
model. For example quark bound states which play the role of QCD pions arise as Goldstone bosons from the spontaneous symmetry breaking generated upon introducing the probe branes giving fundamental degrees of freedom. Recent developments of mesons at finite temperature may be found in [102]. One recent approach generating quark masses dynamically can be found in [103].
Fundamentalism & phenomenology Let us briefly discuss the phenomenological versus fundamental value of AdS/CFT. Although still only a conjecture AdS/CFT has failed no com-parative test so far and it succeeds in describing strong coupling phenomena. The perturbative or geometric understanding on the gravity side can be translated to an understanding of the strongly coupled gauge theory on the other side of the correspondence. In this way AdS/CFT makes it possible to get a qualitative understanding of strong coupling phenomena. At the present level where we do not have an explicit QCD-gravity dual the qualitative understand-ing AdS/CFT supplies us with should be seen as beunderstand-ing complementary to for example lattice data providing exact QCD data but also hiding the inner workings of the strongly coupled theory. In some cases such as for the viscosity bound the quantities involved may even be protected by universality and thus solely depend on the fact that the gauge theory is strongly coupled. If this is the case then AdS/CFT results may even continue to be valid for QCD or the real world. All these results justify the duality at least as a valid phenomenological tool.
Turning around the argument, the phenomenological success of AdS/CFT may be seen as a hint that the gauge gravity correspondence and the principles from which it was derived come indeed close to the principles governing nature. Studying explicit instances of the cor-respondence, for example studying correlators in the D3/D7-setup, could also provide us with a detailed understanding of how the duality works in general and it might even suggest a way to prove AdS/CFT.